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Questions and Answers
What does the pigeonhole principle state in this context?
How many integers can be chosen from the set {1,2,3,...,100} to guarantee at least one pair of multiples?
Which of the following sets is most appropriate to illustrate the pigeonhole principle stated?
What is the maximum number of integers that can be selected without assuring a pair of multiples?
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If 51 integers are chosen, how many unique bases can exist with respect to multiples?
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Study Notes
Pigeonhole Principle
- States that if n items are put into m containers and if n > m, then at least one container must contain more than one item.
- Illustrates a principle of counting and is widely used in combinatorics and probability.
Application to Integers
- Consider the set of integers from 1 to 100.
- All integers can be expressed in the form of a product involving powers of 2 or 5, thus creating distinct "pigeonholes."
- Split the set into pairs: (1,2), (3,6), (4,8), ..., (32,64), ensuring that one number in each pair is a multiple of the other.
Selection of Integers
- If 51 integers are selected from the set of 100 integers, there are only 50 unique pairs (distinct multiples).
- By the Pigeonhole Principle, at least one of these pairs must contain both integers from the selected 51 numbers.
Conclusion
- Thus, among any 51 selected integers from 1 to 100, there will always exist at least two integers where one is a multiple of the other.
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Description
Explore the Pigeonhole Principle through the selection of integers from the set {1, 2, 3, ..., 100}. This quiz examines how choosing 51 integers guarantees that at least two will be multiples of each other. Test your understanding of combinatorial principles and their applications.
